Global convergence of SSM for minimizing a quadratic over a sphere

Authors:
William W. Hager and Soonchul Park

Journal:
Math. Comp. **74** (2005), 1413-1423

MSC (2000):
Primary 90C20, 65F10, 65Y20

DOI:
https://doi.org/10.1090/S0025-5718-04-01731-4

Published electronically:
December 30, 2004

MathSciNet review:
2137009

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Abstract: In an earlier paper [*Minimizing a quadratic over a sphere*, SIAM J. Optim., 12 (2001), 188-208], we presented the sequential subspace method (SSM) for minimizing a quadratic over a sphere. This method generates approximations to a minimizer by carrying out the minimization over a sequence of subspaces that are adjusted after each iterate is computed. We showed in this earlier paper that when the subspace contains a vector obtained by applying one step of Newton's method to the first-order optimality system, SSM is locally, quadratically convergent, even when the original problem is degenerate with multiple solutions and with a singular Jacobian in the optimality system. In this paper, we prove (nonlocal) convergence of SSM to a global minimizer whenever each SSM subspace contains the following three vectors: (i) the current iterate, (ii) the gradient of the cost function evaluated at the current iterate, and (iii) an eigenvector associated with the smallest eigenvalue of the cost function Hessian. For nondegenerate problems, the convergence rate is at least linear when vectors (i)-(iii) are included in the SSM subspace.

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Additional Information

**William W. Hager**

Affiliation:
Department of Mathematics, P.O. Box 118105, University of Florida, Gainesville, Florida 32611-8105

Email:
hager@math.ufl.edu

**Soonchul Park**

Affiliation:
Department of Mathematics, P.O. Box 118105, University of Florida, Gainesville, Florida 32611-8105

Email:
scp@math.ufl.edu

DOI:
https://doi.org/10.1090/S0025-5718-04-01731-4

Keywords:
Quadratic optimization,
quadratic programming,
trust region subproblem,
large-scale optimization,
sparse optimization.

Received by editor(s):
August 12, 2003

Received by editor(s) in revised form:
March 27, 2004

Published electronically:
December 30, 2004

Additional Notes:
This material is based upon work supported by the National Science Foundation under Grant No. 0203270

Article copyright:
© Copyright 2004
American Mathematical Society